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Mirrors > Home > MPE Home > Th. List > canthwdom | Structured version Visualization version GIF version |
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9158, equivalent to canth 7367). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
canthwdom | ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5351 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | ne0i 4335 | . . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅) |
4 | brwdomn0 9603 | . . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) |
6 | 5 | ibi 266 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
7 | relwdom 9600 | . . . . 5 ⊢ Rel ≼* | |
8 | 7 | brrelex2i 5730 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V) |
9 | foeq2 6802 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥)) | |
10 | pweq 4612 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
11 | foeq3 6803 | . . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
13 | 9, 12 | bitrd 278 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
14 | 13 | notbid 317 | . . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴)) |
15 | vex 3467 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | 15 | canth 7367 | . . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥 |
17 | 14, 16 | vtoclg 3534 | . . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
18 | 8, 17 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
19 | 18 | nexdv 1932 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
20 | 6, 19 | pm2.65i 193 | 1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 Vcvv 3463 ∅c0 4323 𝒫 cpw 4598 class class class wbr 5144 –onto→wfo 6542 ≼* cwdom 9598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-wdom 9599 |
This theorem is referenced by: pwdjudom 10248 |
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